6 PS Solutions Econ 367 Kaushik Basu 1. Let us consider Firm 1. Since they are symmetric firms, the analysis will be the same for firm 2. P=100-Q Q= q1 + q2 Profit firm 1 = P*q1 Firm 1 maximizes (100-( q1 + q2)) q1 =100q1-q12-q1q2 Differentiate w/ respect to q1 and set equal to zero to maximize 100-2q1-q2=0 q1=50-q2/2 By symmetry q2=50-q1/2 Solving the system of these two best response functions, the NE is (100/3,100/3) The firms could make more money by producing the monopoly amount and sharing profits. We know that monopoly profits is greater than the Cournot firm's profits because the monopolist could have picked q1*+q2* but did not. In an unlimited capacity state, firm 2 produces more than 20. When limited to 20, firm 1 will plug this amount into its best response function and produce 40. NE (40,20)
2. In a symmetric Bertrand game, the firms always cut their prices down to cost in an attempt to gain more profit. Here cost=0. For any price firm 1 chooses above zero, the other firm will price one cent below and gain the entire market. Then Firm 1 will undercut this new price in order to gain a profit greater than 0. This continues until price=0. NE = (0,0) Price must be an integer: First, we know that with a NE the firms will choose the same number, since if they do not, one will have zero profit, from which it will prefer to deviate. (0,0) is a NE since a unilateral deviation to a higher price will also result in zero profit. (1,1) is a NE since if a firm lowers its price to 0 it will have zero profit. Raising price would also give the deviating firm zero profit. (2,2) is a NE. At (2,2) each firm has one half the market. If a firm unilaterally lowered the price to 1, it would capture the entire market, but the price would have been cut in half, so the total profit the firm faces is the same as if the price were maintained at 2. Again, one firm raising the price in this scenario would result in zero profits for that firm and so it would not deviate in this direction either. Is (3,3) a NE? No, because one firm can lower its price to two and gain the entire market. 2*1 > 3*1/2. From the same type of logic, we assume that we can find no more NE. Solve 3. in the same manner as #1 Firm 1's profit= Px1-cx1
= kx1/(x1+x2)-cx1 Differentiate: k/(x1+x2)-x1k/(x1+x2)2-c=0 x1 = (x2k/c)1/2-x2 By symmetry: x2 = (x1k/c)1/2-x1 The NE is the solution to this system of equations: x1 = x2 = k/4c. We notice that if c is zero, the best response is undefined, since c appears in the denominator. This is because the demand function goes asymptotically to zero along the x axis; it never actually arrives at zero. So if the price is low enough, we can have infinitely many customers, and maximize profit. Bizarre! 4. A strategy p is immune to a strategy q if (p,p) > (q,p) or (p,p) = (q,p) and (p,q) > (q,q). A strategy is evolutionary stable if it is immune to all other strategies. Let us check if the Hawk strategy is evolutionary stable. (H,H)=0 and (D,H)=2. This violates the rule, so Hawk is not immune to Dove and thus cannot be stable. Let us now check Dove. (D,D)=4 and (H,D)=6. This also violates, so there is no evolutionary stable strategy. 5. Let n be the number of days left until the day Akerlof takes the package with him to the friend. Let X be the value of the package, and x the value to Akerlof of Stiglitz having the package each additional day. c is the cost of mailing. P is the salience premium. Then the value of the package being mailed today is nx + X c(1+p) and the value of postponing until tomorrow is (n-1)x + X c. Plugging in our values given in the question, we will procrastinate if 2n + X 5(1+p) < 2(n-1) + X 5. This reduces to p>2/5, meaning p must be at least 2/5 for procrastination to occur. Now we wish to know how long the vacation must be for Akerlof to mail the package today. This implies: 2n + X 5(1+2) > 2(n-v) + X 5. This simplifies to v>5, which implies the vacation must be at least 5 days long for the package to be mailed away. 6. An equilibrium selection norm arises when there are more than one Nash equilibrium in a game. The question is, how do we know at which equilibrium we arrive? Social norms is one possibility. The example from class is driving two equilibria are everyone drives on the right side or everyone drives on the left. Have fun thinking of others!